Abstract
The metric tensor having been introduced, the notions of orthogonality and local parallelism may then be introduced by applying the classical Euclidean laws to infinitesimal triangles. For this purpose, the laws of similar triangles must be adopted ab initio as postulates, and the ‘postulate of parallels’ must be excluded, a procedure that is reasonable as far as physics is concerned both because the laws of similar triangles correspond immediately to the intuition of experience and because experience is always limited to finite regions. By defining the right angle as the angle of intersection of two lines that makes all four intersection angles equal, and by guaranteeing its uniqueness through further axiomatic refinements on the comparison of angles by means of the notions ‘greater than’ and ‘less than’ as well as ‘equality’, one may then derive the Pythagorean theorem in the well-known manner.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Mathematicians sometimes call the manifold pseudo-Riemannian when the metric is not positive definite.
- 2.
Mathematicians sometimes refer to Riemannian manifolds for which the metric tensor has one eigenvalue of one sign while all the others have opposite sign as Lorentzian manifolds.
- 3.
We confine our attention in these lectures to diffeomorphisms that may be connected continuously to the identity.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
DeWitt, B., Christensen, S.M. (2011). Riemannian Manifolds. In: Christensen, S. (eds) Bryce DeWitt's Lectures on Gravitation. Lecture Notes in Physics, vol 826. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36911-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-36911-0_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36909-7
Online ISBN: 978-3-540-36911-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)